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'Maxwell's equations'
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| Title of object: |
Maxwell's equations |
| Canonical Name: |
MaxwellsEquations |
| Type: |
Definition |
| Created on: |
2008-02-26 14:20:17 |
| Modified on: |
2008-03-14 16:37:26 |
| Classification: |
msc:35Q60 |
| Defines: |
Faraday's Law, Ampere's Law, Gauss' Law of Electrostatics, Gauss' Law of Magnetostatics |
Preamble:
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Content:
Maxwell's equations are a set of four partial differential equations first combined by James Clerk Maxwell. They may also be written as integral equations. Two other important equations, the electromagnetic wave equation and the equation of conservation of charge, may be derived from them.
\subsection{Notation}
As this article considers merely the mathematical aspects of the equations, natural units have been used throughout. For their use in physics see any electromagnetism textbook.
\[
\mathbf{E} = \mbox{Electric field strength}
\]
\[
\mathbf{B} = \mbox{Magnetic flux density}
\]
\subsection{Gauss' Law of Electrostatics}
Differential form
\[
\nabla \cdot \mathbf{E} = 0
\]
Integral form
\[
\oint_S \mathbf{E} \cdot \mathrm{d}\mathbf{S} = 0
\]
\subsection{Gauss' Law of Magnetostatics}
\[
\nabla \cdot \mathbf{B} = 0
\]
\[
\oint_S \mathbf{B} \cdot \mathrm{d}\mathbf{S} = 0
\]
This law can be interpreted as a statement of the non-existence of magnetic monopoles, a fact confirmed by all experiments to date.
\subsection{Faraday's Law}
Differential form
\[
\nabla \times \mathbf{E} = -\frac{ \partial \mathbf{B}}{\partial t}
\]
\subsection{Amp\`ere's Law}
Differential form
\[
\nabla \times \mathbf{B} = - \mu_0 \epsilon_0 \frac{ \partial \mathbf{E}}{\partial t}
\]
Integral form
\subsection{Properties of Maxwell's Equations}
These four equations together have several interesting properties:
\begin{itemize}
\item Lorentz invariance
\item Gauge invariance
\item Derivation from an appropriate Lagrangian
\item In natural units, the equations are symmetric in $\mathbf{E}$ and $\mathbf{B}$.
\end{itemize} |
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